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| Mirrors > Home > MPE Home > Th. List > ixxss1 | Structured version Visualization version GIF version | ||
| Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.) |
| Ref | Expression |
|---|---|
| ixx.1 | ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) |
| ixxss1.2 | ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑆𝑦)}) |
| ixxss1.3 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤)) |
| Ref | Expression |
|---|---|
| ixxss1 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ixxss1.2 | . . . . . . . 8 ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑆𝑦)}) | |
| 2 | 1 | elixx3g 13400 | . . . . . . 7 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) ↔ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) ∧ (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶))) |
| 3 | 2 | simplbi 497 | . . . . . 6 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*)) |
| 4 | 3 | adantl 481 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*)) |
| 5 | 4 | simp3d 1145 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤 ∈ ℝ*) |
| 6 | simplr 769 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴𝑊𝐵) | |
| 7 | 2 | simprbi 496 | . . . . . . 7 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) → (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶)) |
| 8 | 7 | adantl 481 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶)) |
| 9 | 8 | simpld 494 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐵𝑇𝑤) |
| 10 | simpll 767 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴 ∈ ℝ*) | |
| 11 | 4 | simp1d 1143 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐵 ∈ ℝ*) |
| 12 | ixxss1.3 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤)) | |
| 13 | 10, 11, 5, 12 | syl3anc 1373 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤)) |
| 14 | 6, 9, 13 | mp2and 699 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴𝑅𝑤) |
| 15 | 8 | simprd 495 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤𝑆𝐶) |
| 16 | 4 | simp2d 1144 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐶 ∈ ℝ*) |
| 17 | ixx.1 | . . . . . 6 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
| 18 | 17 | elixx1 13396 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶))) |
| 19 | 10, 16, 18 | syl2anc 584 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶))) |
| 20 | 5, 14, 15, 19 | mpbir3and 1343 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤 ∈ (𝐴𝑂𝐶)) |
| 21 | 20 | ex 412 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) → (𝑤 ∈ (𝐵𝑃𝐶) → 𝑤 ∈ (𝐴𝑂𝐶))) |
| 22 | 21 | ssrdv 3989 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {crab 3436 ⊆ wss 3951 class class class wbr 5143 (class class class)co 7431 ∈ cmpo 7433 ℝ*cxr 11294 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-xr 11299 |
| This theorem is referenced by: iooss1 13422 limsupgord 15508 pnfnei 23228 dvfsumrlimge0 26071 dvfsumrlim2 26073 tanord1 26579 rlimcnp 27008 rlimcnp2 27009 dchrisum0lem2a 27561 pntleml 27655 pnt 27658 liminfgord 45769 |
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